Symmetric group:S3
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Contents
Definition
Verbal definitions
The symmetric group can be defined in the following equivalent ways:
- It is the symmetric group on a set of three elements, viz., the group of all permutations of a three-element set. In particular, it is a symmetric group of prime degree and symmetric group of prime power degree.
- It is the dihedral group of order six (degree three), viz., the group of (not necessarily orientation-preserving) symmetries of the equilateral triangle.
- It is the special linear group of degree two over the field of two elements. It turns out that, because of the nature of the prime two, it is also the projective special linear group of degree two , the general linear group of degree two , and the projective general linear group of degree two .
- It is the general affine group of degree one over the field of three elements, i.e., (sometimes also written as ).
- It is the general semilinear group of degree one over the field of four elements, i.e., .
- It is the von Dyck group with parameters , and in particular, is a Coxeter group. In particular, it has the presentation (where denotes the identity element):
.
In the Coxeter language, this is written as:
.
Multiplication table
We portray elements as permutations on the set using the cycle decomposition. The row element is multiplied on the left and the column element on the right, with the assumption of functions written on the left. This means that the column element is applied first and the row element is applied next.
Element | ||||||
---|---|---|---|---|---|---|
If we used the opposite convention (i.e., functions written on the right), the row element is to be multiplied on the right and the column element on the left.
Here is the multiplication table where we use the one-line notation for permutations, where, as in the previous multiplication table, the column permutation is applied first and then the row permutation. Thus, with the left action convention, the row element is multiplied on the left and the column element on the right:
Element | 123 | 213 | 132 | 321 | 231 | 312 |
---|---|---|---|---|---|---|
123 | 123 | 213 | 132 | 321 | 231 | 312 |
213 | 213 | 123 | 231 | 312 | 132 | 321 |
132 | 132 | 312 | 123 | 231 | 321 | 213 |
321 | 321 | 231 | 312 | 123 | 213 | 132 |
231 | 231 | 321 | 213 | 132 | 312 | 123 |
312 | 312 | 132 | 321 | 213 | 123 | 231 |
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Families
The symmetric group on three elements is part of some important families:
Generic name for family member | Definition | Parametrization of family | Parameter value(s) for this member | Other members | Comments |
---|---|---|---|---|---|
symmetric group on finite set | group of all permutations on a finite set | by a nonnegative integer , denoting size of set acted on | , so the group is | click here for a list | |
Coxeter group | has a presentation of a particular form | Coxeter matrix describing the presentation | click here for a list | symmetric groups on finite sets are Coxeter groups | |
dihedral group | semidirect product of a cyclic group and a two-element group acting via the inverse map | by a positive integer that's half the order | , so the group is | click here for a list | |
general affine group | semidirect product of a vector space over a field with the general linear group acting on that vector space | name of field , degree (i.e., dimension of vector space). For a finite field, we may also write the group as where is the size of the field | field:F3 (size ), degree one, so the group is or | click here for a list | |
general linear group | general linear group of finite degree over a finite field | name of field , degree . may be replaced by its size in case of a finite field. | field:F2 (size ), degree two, so the group is or | click here for a list | |
projective general linear group | projective general linear group of finite degree over a finite field | name of field , degree . may be replaced by its size in case of a finite field. | field:F2 (size ), degree two, so the group is or | click here for a list | |
special linear group | special linear group of finite degree over a finite field | name of field , degree . may be replaced by its size in case of a finite field. | field:F2 (size ), degree two, so the group is or | click here for a list | |
projective special linear group | projective special linear group of finite degree over a finite field | name of field , degree . may be replaced by its size in case of a finite field. | field:F2 (size ), degree two, so the group is or | click here for a list | |
general semilinear group | semidirect product of general linear group and automorphism group of base field | name of field , degree . may be replaced by its size in case of a finite field. | field:F4 (size ), degree one, so the group is | click here for a list |
Elements
Further information: Element structure of symmetric group:S3
Conjugacy class structure
As for any symmetric group, cycle type determines conjugacy class. The cycle types, in turn, are parametrized by the unordered integer partitions of . The conjugacy classes are described below.
Partition | Partition in grouped form | Verbal description of cycle type | Elements with the cycle type in cycle decomposition notation | Elements with the cycle type in one-line notation | Size of conjugacy class | Formula for size | Even or odd? If even, splits? If splits, real in alternating group? | Element order | Formula calculating element order |
---|---|---|---|---|---|---|---|---|---|
1 + 1 + 1 | 1 (3 times) | three fixed points | -- the identity element | 123 | 1 | even; no | 1 | ||
2 + 1 | 2 (1 time), 1 (1 time) | transposition in symmetric group:S3: one 2-cycle, one fixed point | , , | 213, 321, 132 | 3 | odd | 2 | ||
3 | 3 (1 time) | 3-cycle in symmetric group:S3: one 3-cycle | , | 231, 312 | 2 | even; yes; no | 3 | ||
Total (3 rows -- 3 being the number of unordered integer partitions of 3) | -- | -- | -- | -- | 6 (equals 3!, the size of the symmetric group) | -- | odd: 3 even;no: 1 even; yes; no: 2 |
order 1: 1, order 2: 3, order 3: 2 | -- |
This group is one of three finite groups with the property that any two elements of the same order are conjugate. The other two are the cyclic group of order two and the trivial group.
For an interpretation of the conjugacy class structure based on the other equivalent definitions of the group, visit Element structure of symmetric group:S3#Conjugacy class structure.
Automorphism class structure
The classification of elements upto automorphism is the same as that upto conjugation; this is because the symmetric group on three elements is a complete group: a centerless group where every automorphism is inner.
Arithmetic functions
Basic arithmetic functions
Arithmetic functions of an element-counting nature
Arithmetic functions of a subgroup-counting nature
Lists of numerical invariants
List | Value | Explanation/comment |
---|---|---|
conjugacy class sizes | 1,2,3 | See cycle type determines conjugacy class, element structure of symmetric group:S3, element structure of symmetric groups |
order statistics | ||
degrees of irreducible representations | 1,1,2 | See linear representation theory of symmetric group:S3, linear representation theory of symmetric groups |
orders of subgroups | 1,2,2,2,3,6 | See subgroup structure of symmetric group:S3 |
Endomorphisms
Further information: Endomorphism structure of symmetric group:S3
Automorphisms
Since is a complete group, it is isomorphic to its automorphism group, where each element of acts on by conjugation. In fact, for , the symmetric group is a complete group. Further information: Symmetric groups on finite sets are complete
Group properties
Important properties
Property | Satisfied? | Explanation | Comment |
---|---|---|---|
Abelian group | No | and don't commute | Smallest non-abelian group |
Nilpotent group | No | Centerless: The center is trivial | Smallest non-nilpotent group |
Metacyclic group | Yes | Cyclic normal subgroup of order three, cyclic quotient of order two | |
Supersolvable group | Yes | Metacyclic implies supersolvable | |
Solvable group | Yes | Metacyclic implies solvable |
Other properties
Property | Satisfied? | Explanation | Comment |
---|---|---|---|
T-group | Yes | ||
Monolithic group | Yes | Unique minimal normal subgroup of order three | |
One-headed group | Yes | Unique maximal normal subgroup of order three | |
Jordan-unique group | Yes | There is a unique composition series | |
SC-group | Yes | Every subgroup of it is a C-group | C-group means that every subgroup is permutably complemented |
Rational-representation group | Yes | Symmetric groups are rational-representation | |
Rational group | Yes | Symmetric groups are rational | Also see classification of rational dihedral groups |
Ambivalent group | Yes | Symmetric groups are ambivalent | |
Complete group | Yes | Symmetric groups are complete, except degrees | |
Frobenius group | Yes | Frobenius kernel is alternating group, complement is any subgroup of order two. | Frobenius group on account of being . |
Camina group | Yes | ||
Z-group | Yes | Both the 2-Sylow subgroup (S2 in S3) and the 3-Sylow subgroup (A3 in S3) are cyclic. | |
Schur-trivial group | Yes | Schur multiplier of Z-group is trivial |
Subgroups
Further information: Subgroup structure of symmetric group:S3
Quick summary
Item | Value |
---|---|
Number of subgroups | 6 Compared with : 1,2,6,30,156,1455,11300, 151221 |
Number of conjugacy classes of subgroups | 4 Compared with : 1,2,4,11,19,56,96,296,554,1593 |
Number of automorphism classes of subgroups | 4 Compared with : 1,2,4,11,19,37,96,296,554,1593 |
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems | 2-Sylow: cyclic group:Z2, Sylow number is 3, fusion system is the trivial one 3-Sylow: cyclic group:Z3, Sylow number is 1, fusion system is non-inner fusion system for cyclic group:Z3 |
Hall subgroups | Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups. Interestingly, all subgroups are Hall subgroups, because the order is a square-free number |
maximal subgroups | maximal subgroups have order 2 (S2 in S3) and 3 (A3 in S3). |
normal subgroups | There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3. |
Table classifying subgroups up to automorphisms
For more information on each automorphism type, follow the link.
Automorphism class of subgroups | List of all subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes (=1 iff automorph-conjugate subgroup) | Size of each conjugacy class (=1 iff normal subgroup) | Total number of subgroups (=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Note |
---|---|---|---|---|---|---|---|---|---|
trivial subgroup | trivial group | 1 | 6 | 1 | 1 | 1 | symmetric group:S3 | trivial | |
S2 in S3 | cyclic group:Z2 | 2 | 3 | 1 | 3 | 3 | -- | 2-Sylow | |
A3 in S3 | cyclic group:Z3 | 3 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 3-Sylow | |
whole group | |
symmetric group:S3 | 6 | 1 | 1 | 1 | 1 | trivial group | |
Total (4 rows) | -- | -- | -- | -- | 4 | -- | 6 | -- | -- |
Subgroup-defining functions and associated quotient-defining functions
Linear representation theory
Further information: Linear representation theory of symmetric group:S3
Summary
Item | Value |
---|---|
Degrees of irreducible representations over a splitting field (and in particular over ) | 1,1,2 maximum: 2, lcm: 2, number: 3 sum of squares: 6, quasirandom degree: 1 |
Schur index values of irreducible representations | 1,1,1 |
Smallest ring of realization for all irreducible representations (characteristic zero) | |
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) | (hence, it is a rational representation group) |
Condition for being a splitting field for this group | Any field of characteristic not two or three is a splitting field. In particular, and are splitting fields. |
Minimal splitting field in characteristic | The prime field |
Smallest size splitting field | field:F5, i.e., the field of five elements. |
Character table
Representation/Conjugacy class representative | (identity element) -- size 1 | (3-cycle) -- size 2 | (2-transposition) -- size 3 |
---|---|---|---|
Trivial representation | 1 | 1 | 1 |
Sign representation | 1 | 1 | -1 |
Standard representation | 2 | -1 | 0 |
Distinguishing features
Smallest of its kind
- This is the unique non-abelian group of smallest order. All groups of order up to , and all other groups of order , are abelian.
- This is the unique non-nilpotent group of smallest order. All groups of order up to , and all other groups of order , are nilpotent.
- This is the unique smallest nontrivial complete group.
GAP implementation
Group ID
This finite group has order 6 and has ID 1 among the groups of order 6 in GAP's SmallGroup library. For context, there are groups of order 6. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(6,1)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(6,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [6,1]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Other descriptions
Description | Functions used |
---|---|
SymmetricGroup(3) | SymmetricGroup |
DihedralGroup(6) | DihedralGroup |
GL(2,2) | GL |